Connected sums of unstabilized Heegaard splittings are unstabilized

Abstract

be closed, orientable 3-manifolds. Let H i denote a Heegaard surface in M i . We prove that if H 1 #H 2 comes from stabilizing a lower genus splitting of M 1 #M 2 then one of H 1 or H 2 comes from stabilizing a lower genus splitting. This answers a question of C Gordon [9, Problem 3.91]. We also show that every unstabilized Heegaard splitting has a unique expression as the connected sum of Heegaard splittings of prime 3-manifolds.

Mapping Class Groups of Heegaard Splittings

“…Just as strongly irreducible Heegaard surfaces define minimal surfaces of index one, such loops of embeddings should determine minimal surfaces of index two. See Bachman [2] and Hass, Thompson, Thurston [12] for further developments of similar ideas.…”

Generating tuples of free products

“…We now define the graph Γ [T ] to be the directed graph whose vertex set is V Γ [T ] := Ω [T ] . Two vertices ω 1 and ω 2 are further joined by a directed edge from ω 1 to ω 2 (which we denote by ω 1 → ω 2 ) if there are representatives B 1 and B 2 of ω 1 and ω 2 such that B 2 is the core of the A-graph obtained from B 1 by an elementary fold of type F 1 or F 4.…”

“…The original question asked whether the connected sum of irreducible Heegaard splittings is necessarily irreducible. Affirmative answers to this question have recently been announced by D. Bachman [1] and Ruifeng Qiu [13].…”

Topologically minimal surfaces versus self-amalgamated Heegaard surfaces

“…If a weakly reducible surface is topologically minimal, the topological index is at least 2. Index 2 topologically minimal surface are called critical surfaces which are also defined by Bachman, see [1][2][3][4] .…”